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The interaction between nanoparticles and ultrashort laser pulses holds great interest in laser nanomedicine, introducing such possibilities as selective cell targeting to create highly localized cell damage. Two models are studied to describe the laser pulse interaction with nanoparticles in the femtosecond, picosecond, and nanosecond regimes. The first is a two-temperature model using two coupled diffusion equations: one describing the heat conduction of electrons, and the other that of the lattice. The second model is a one-temperature model utilizing a heat diffusion equation for the phonon subsystem and applying a uniform heating approximation throughout the particle volume. A comparison of the two modeling strategies shows that the two-temperature model gives a good approximation for the femtosecond mode, but fails to accurately describe the laser heating for longer pulses. On the contrary, the simpler one-temperature model provides an adequate description of the laser heating of nanoparticles in the femtosecond, picosecond, and nanosecond modes.

The application of ultrashort laser pulse thermal-based killing of
abnormal cells (e.g., cancer cells) targeted with absorbing nanoparticles (e.g.,
solid gold nanospheres, nanoshells, or nanorods) is becoming an extensive area
of research [

Traditional knowledge of laser-nanoparticle interactions has necessitated
specialized models for each case, dependent upon the laser pulse duration.
Previous research has recognized the existence of these conditions and
dual-temperature models for the ultrashort laser pulse mode calculating
electron, and lattice subsystem temperatures are readily available (see, e.g.,
[

Ultrashort pulses, specifically those in the femtosecond and picosecond ranges, impose several challenges in modeling material response. Free electrons with minimal capacity for heat are the first to absorb energy, rapidly attaining high temperatures, and transferring thermal energy to the material lattice. These processes do not occur instantaneously: time must be allowed for the cooling of the electrons and the heating of the lattice. In our work, electron cooling and lattice heating have time delays on the order of femtoseconds and picoseconds, respectively. Ultrashort laser pulses end before the transfer of energy to the lattice is complete, requiring two-temperature models in order to describe the further conversion of energy from electron excitation to heat within the lattice system.

In the present paper, we demonstrate that a simpler, one-temperature model (OTM) utilizing the uniform heating approximation is appropriate for understanding ultrashort laser pulse interactions with metal nanoparticles. The approximation may be used in this situation due to the extremely small size of nanoparticles in comparison to the wavelength of laser radiation. Using this idea, the time delay between the electron and lattice interactions will be bypassed, and the simpler model will ultimately yield results similar to the two-temperature model (TTM). Comparative simulations of the two modeling approaches are performed in this paper to confirm OTM as an appropriate approximation for nanoparticle heating in the femtosecond, picosecond, and nanosecond regimes, thus providing an effective modeling method for further nanomedicine research to explore.

During the interaction of a laser pulse of intensity

The power density of energy generation in the particle

Heat exchange between the nanoparticle surface and the surrounding medium
is rapid, and heat loss becomes substantial for relatively long laser pulses.
Assuming that the heat lost from the surface of nanoparticle occurs only due to
heat diffusion into surrounding medium, the energy flux density

After integration over the volume for the spherically symmetric case and
transition to a uniform temperature over the particle volume, the equation
which describes the kinetics of laser heating of the nanoparticle and results
from (

In
TTM, the temperature relaxation in time and sample depth can be modeled by
two-coupled diffusion equations: one describing the heat conduction of
electrons and the other that in the lattice. Both equations are connected by a
term that is proportional to the electron-phonon coupling constant

Input parameters used for simulations in both models.

Parameters | Magnitude and units | Reference |
---|---|---|

Laser pulse shape | ||

Energy density | ||

Volume density of the gold | [ | |

Electron heat capacity (gold) | [ | |

Electron-phonon coupling constant for gold | [ | |

Gold specific heat | ||

Specific heat of water | ||

Radius of gold nanoparticle | ||

Absorption efficiency of gold nanoparticle | [ | |

Absorption coefficient of gold nanoparticle | [ | |

Power exponent | ||

Thermal conductivity of water |

Comparative simulations
using the models described above have been performed for the laser heating of a
gold nanoparticle with radius

TTM and OTM have been
solved numerically to predict the time dependence of the electron and lattice
temperatures in the femtosecond mode when the laser pulse duration is shorter
than the electron thermalization and cooling times, ^{2} and pulse duration of ^{2} to provide the cell lethality during a single laser
pulse. The results of the simulations for the heating of a gold nanoparticle by
a femtosecond laser pulse are shown in Figures

(a) Electron (solid curve) and lattice (dashed curve) temperature evolutions on the femtosecond time scale for a gold nanoparticle predicted by TTM. (b) Evolution of the nanoparticle temperature (dashed curve) after the femtosecond laser pulse, predicted by OTM, and laser pulse shape (solid curve).

Comparison of nanoparticle temperature evolutions after the 60 femtoseconds laser pulse-predicted TTM (solid curve) and one-temperature model (dashed curve).

As follows from Figure

(a) Electron temperature relaxation on the picosecond time scale. (b) Temperature time distributions for a gold nanoparticle predicted by OTM (solid curve) and TTM (dashed curve) after the 60 picoseconds laser pulse.

The slow rate of electron
heat diffusion into the phonon subsystem on the femtosecond time scale results
in a delay of about 100 femtoseconds in the heating of the bulk sample (dashed curve in Figure ^{2}.

The results of heating the
gold nanoparticle obtained by OTM are demonstrated in Figure

In this mode, the
constants

Opposite to TTM, OTM
describes the picosecond heating kinetics very well. A typical time evolution
of the particle temperature predicted by OTM on the picosecond time scale is
displayed in Figure

For laser heating of
metal nanoparticles in the nanosecond regime, the characteristic lattice
heating time

Sample calculations have
been carried out using OTM for gold nanoparticles with radii ^{2} and pulse duration ^{2} is comparable to the laser fluence currently used in the
photothermolysis of cancer cells [

Thermophysical characteristics of the gold particle and surrounding biological tissue.

Material | Specific heat | Interval of | Thermal conductivity μ_{0} (W/m K) | Thermal diffusivity |
---|---|---|---|---|

Gold | 129 | 273–373 | 318 | |

Water | 273–373 | |||

Human prostate | 3740 | 310 | 0.529 | |

Blood | 3645–3897 | 273–373 | 0.48–0.6 | |

Fat | 2975 | 273–373 | 0.185–0.233 | |

Tumor | 3160 | 310 | 0.561 | |

Skin | 273–373 | 0.210–0.410 |

Kinetics of heating and cooling of a gold nanoparticle by a nanosecond laser
pulse of energy density 10 mJ/cm^{2} and duration 8 nanoseconds.
Calculations have been made by using OTM. (a) Illustration of the time dynamics
of laser heating of a 30-nm gold particle in different biological media: fat
(dashed-dotted curve), blood (dashed curve), tumor (solid curve), and prostate
(dotted curve). (b) Results of thermal calculations for a 35-nm gold particle:
heating and cooling in the water surrounding medium at different heat transfer
rates

It follows from these
calculations that during the laser pulse duration the transfer of heat from the
nanoparticle into the surrounding media is slight, and the particle rapidly
reaches a high temperature. The heating rate is about 10^{12} Ks^{-1}. The temperature of the particle continues to rise even after the end of the
laser pulse. The highest temperature, 770 K, for a given laser pulse fluence is
observed for the heating time of 13.5 nanoseconds, when the laser pulse has
already degraded (see Figure

We have also examined the
effect of different biological surroundings on the laser heating dynamics of 30 nm gold particles. Four biomedia were used: namely, blood, human prostate,
tumor, and fat. Results of computer simulations of the time-temperature
profiles of gold nanoparticles in various biological media, performed by using
OTM, are plotted in Figure

The temperature dynamics
of the particle is sensitive to the power exponent

It is interesting to
investigate the effect of the particle’s radius on the temperature dynamics of
the nanoparticle heated by the nanosecond laser radiation in the biological
surroundings. There are two competitive factors here. On one hand, according to
the Mie diffraction theory, the absorption efficiency of the gold nanoparticle
drops with the decreasing size of the particle. On the other hand, the heating
rate increases for smaller particles as follows from (

Nondimensional (a.u.) absorption efficiency ^{2} and duration 8 nanoseconds.

The comparative analysis of OTM and TTM for heating of a metal nanoparticle in the femtosecond, picosecond, and nanosecond regimes has shown that

in the femtosecond mode, the thermal equilibrium among the excited electrons is established within the first 175 femtoesonds, long after the end of the laser pulse duration;

the electrons remain in the thermal equilibrium state up to 1 picosecond;

both models demonstrate the same scenario in the heating kinetics of a metal nanoparticle by a femtosecond laser pulse: about a 100-femtosecond time delay in the heating of the particle is observed, until reaching a maximum lattice temperature and saturation in temperature curves after 175 femtoeconds;

the electron cooling time due to coupling to the lattice is about 10 picoecond, which imposes an upper time limit for TTM application;

TTM gives a very good approximation for the femtosecomd mode while an electron temperature exists, but it fails to describe the laser heating of nanoparticles for longer pulse durations in the picosecond and nanosecond regimes;

OTM shows that the heat lost from the surface of the nanoparticle into the surrounding medium becomes noticeable after 200 picoseconds;

the heating of a metal nanoparticle by a nanosecond laser pulse in fat provides higher particle overheating than in blood, prostate, and water as surrounding media due to the thermally isolating property of the fat;

the optical properties of the nanoparticle have a much stronger effect on the heating dynamics in the nanosecond mode than the thermal effects when the radius of the particle is less than 35 nm. For larger particles, the thermal processes dominate the optical properties, and the temperature curve is determined by the balance between heating of the nanoparticle and energy losses from the surface of the particle due to heat diffusion into the surrounding biological medium.

This work has been supported by the Lilly Foundation Grant AA0000010.